We draw letters (a-z) independently and uniformly at random from the alphabet with replacement. If we draw a collection of letters, then we say that we can write a word when the collections contains all the letters needed to spell the word. For example, if we draw ‘a’, ‘c’ and ‘t’, then this collection writes the word ‘cat’, but also ‘act’, ‘at’ and ‘a’.
Let $N$ be the random variable that denotes the number of letters we needed to draw before we can write ‘rowel’ for the first time. Find $E[N]$ and $\text{Var}[N]$.
Could someone help me by showing me how to approach this question?
At first you have probability $\frac5{26}$ on each try to draw a letter you need. Once you’ve succeeded, you have probability $\frac4{26}$ on each try to draw another letter you still need, and so on. Thus $N$ is the sum of $5$ independently geometrically distributed variables, whose expectations and variances you can add to find the expectation and variance of $N$.
This is a truncated version of the coupon collector’s problem. You’ll find a lot of related questions under the corresponding tag that was added to your question.